Analysis Seminar

Speaker: Javad Rastegari Koopaei (Western)

"Norm inequalities for the Fourier transform on the unit circle"

Time: 14:30

Room: MC 107

The Fourier transform, $\mathcal{F}$, enjoys certain boundedness properties as a map between various normed spaces. Typical examples are : $\|\widehat{f}\|_{L^{\infty}} \leq \|f\|_{L^1}$, $\|\widehat{f}\|_{L^2} = \|f\|_{L^2}$ (Plancherel theorem) and $\|\widehat{f}\|_{L^{p'}} \leq C \|f\|_{L^p}$ , (Hausdorff-Young inequality with $1< p \leq 2$ and $1/p + 1/p' = 1$).

This talk starts with a brief history of Fourier inequalities in weighted $L^p$ spaces and weighted Lorentz spaces. The Lorentz norm with weight $w$ is defined as $\|f\|_{\Lambda_w^p} = \|f^*\|_{L_w^p}$ where $f^*$ is decreasing rearrangement of $f$.

Then I will focus on our recent joint work with G. Sinnamon on norm inequalities for Fourier series. I will present relationship between weight functions $u(t), w(t)$ and exponents $(p,q)$ that is sufficient/necessary for $\|\widehat{f}\|_{\Lambda_u^q}\leq C\|f\|_{\Lambda_w^p}$.

An immediate consequence is some new results on boundedness of $\mathcal{F} : L_w^p (\mathbb{T}) \longrightarrow L_u^q(\mathbb{Z})$ where $u[n]$ and $w(t)$ are weight functions on $\mathbb{Z}$ and the unit circle $\mathbb{T}$ respectively.

Comprehensive Exam Presentation

Speaker: Chandra Rajamani (Western)

"PhD Comprehensive Presentation"

Time: 13:30

Room: MC 108

Let M be a symplectic manifold. It has a group of structure preserving automorphisms called the Symplectomorphism group. This group has a lie subgroup called the Hamiltonian group. It is known that the number of conjugacy classes of maximal tori in Ham by Symp is finite for 4 dimensional M. This result also holds in general when the torus has half the dimension of M. We hope to generalize this result to Symplectic orbifolds. This result leads to implications on conjugacy classes of maximal tori in the Contactomorphism group of a compact contact manifold.

Algebra Seminar

Speaker: Claudio Quadrelli (Western and Milano-Bicocca)

"Galois pro-$p$ groups on a diet of roots of the field"

Time: 14:30

Room: MC 107

For a field $F$ containing a primitive $p$-root of $1$, let $G$ be the Galois group of the maximal $p$-extension $F(p)$ of $F$. The group $G$ might become very fat -- i.e., the size of its open subgroups might increase arbitrarily. This does not happen (namely, the size of its open subgroups is always the same) precisely when $F$ needs to eat only the roots of $p$-power index of its elements to reach $F(p)$. In this case we may compute explicitly the structure of $G$.

PhD Thesis Defence

Speaker: Claudio Quadrelli (Western and Milan)

"The Structure of Absolute Galois Groups"

Time: 09:00

Room: MC 108

Analysis Seminar

Speaker: Edward Bierstone (University of Toronto)

"Hsiang-Pati coordinates"

Time: 14:30

Room: MC 107

Given a complex analytic (or algebraic) variety X, can we find a resolution of singularities p: Y -> X such that the pulled-back cotangent sheaf is generated by differential monomials in suitable coordinates at every point of Y ("Hsiang-Pati coordinates")? The answer is "yes" in dimension up to 3. It was previously known for surfaces X with isolated singularities (Hsiang- Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety X. In the case of a surface X, Hsiang and Pati used this to prove that the intersection cohomology of X (with the middle perversity) equals the $L^2$ cohomology of the smooth part of X (Cheeger-Goresky- Macpherson conjecture). Existence of Hsiang-Pati coordinates is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of p. (Joint work with Andre Belotto, Vincent Grandjean and Pierre Milman.)

Colloquium

Speaker: Rene Schoof (University of Rome, Tor Vergata)

"Elliptic curves and modular curves"

Time: 15:30

Room: MC 108

 In 1972 J.-P. Serre proved an important theorem concerning the Galois action on the torsion points of elliptic curves over number fields. We describe a conjectural uniform version of this theorem and its relation to rational points on modular curves.

Analysis Seminar

Speaker: Vitali Vougalter (University of Toronto)

"On the solvability in the sense of sequences for some non Fredholm operators"

Time: 14:30

Room: MC 107

We investigate solvability of certain linear nonhomogeneous elliptic equations and establish that under reasonable technical conditions the convergence in $L^2(R^d)$ of their right sides yields the existence and the convergence in $H^2(R^d)$ of the solutions. The problems involve the sums of second order differential operators without Fredholm property and we apply the methods of spectral and scattering theory for Schroedinger type operators.